Chromatic classes of 2-connected (n, n + 3)-graphs with at least two triangles

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G ~ H, if P(G) = P(H). A graph G is chromatically unique if G ? H for any graph H such that H ~ G. Let H denote the class of 2-connected graphs with n vertices and n + 3 edges which contain at least two triangles. It follows that if G ? H and H ~ G, then H ? H. In this paper, we determine all equivalence classes in H under the equivalence relation ~ and characterize the structure of the graphs in each class. As a by-product of these, we obtain various new families of chromatically equivalent graphs and chromatically unique graphs.